Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian

Abstract

In this paper we initiate the study of $2$nd order variational problems in $L^{\infty }$, seeking to minimise the $L^{\infty }$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. Given $\mathrm{H}\in C^1({\mathbb{R}}_s^{n\times n})$, for the functional \[ \label{1} \mathrm{E}_\infty(u,\mathcal{O})\, =\, \big\| \mathrm{H}\big(\mathrm{D}^2 u\big) \big\|_{L^\infty(\mathcal{O})}, \ \ \ u\in W^{2,\infty}(Ω),\ \mathcal{O}\subseteq Ω, \tag{1} \] the associated equation is the fully nonlinear 3rd order PDE \[ \label{2} \mathrm{A}^2_\infty u\, :=\,\big(\mathrm{H}_X\big(\mathrm{D}^2u\big)\big)^{\otimes 3}:\big(\mathrm{D}^3u\big)^{\otimes 2}\, =\,0. \tag{2} \] Special cases arise when $\mathrm{H}$ is the Euclidean length of either the full hessian or of the Laplacian, leading to the $\infty$-Polylaplacian and the $\infty$-Bilaplacian respectively. We establish several results for \eqref{1} and \eqref{2}, including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

arXiv commentJournal: Advances in Calculus of Variations, 31 pages, 13 figures

References in corpus (6)

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Cited by in corpus (7)

• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^{\infty }$
• A New Characterisation of $\infty$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^{\infty }$
• A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in $L^{\infty }$
• Solutions of Vectorial Hamilton-Jacobi Equations are Rank-One Absolute Minimisers in $L^{\infty }$
• On the numerical approximation of $p$-Biharmonic and $\infty$-Biharmonic functions
• Explicit $\infty$-harmonic functions in high dimensions
• The Eigenvalue Problem for the $\infty$-Bilaplacian